In the equations 3.59 it is also assumed that the total atomic density is unchanged by the presence of the laser field. This can be a good assumption for hot vapours but not for cold atoms.
The termLe[ne] is a linear operator that depends on the nature of the excited states transport processes.
The first terms on the right of equations 3.59 is the rate of local excitation αI
~ω and the spatial distribution of the excited state densityne is therefore given by :
ne(r⊥, t) =
At this point it is possible to insert in this formula a spatially dependent intensity profile and to computene(r⊥).
Considering a temporally constant field, i.e a delta function in space I =I0δ(r0) : ne(r) = α(I0)I0
~ωπ˜vr Z ∞
0
dξe−Γr˜vξe−ξ2 . (3.62) Thus, defining r=|r−r0|the Green function for a steady state spatial response is :
R(r,r0; Γ) = 1 π˜vr
Z ∞ 0
dξe−Γr˜vξe−ξ2 , (3.63) that can be Fourier transformed in order to be expressed in function of the wavevector k:
The nonlinear response function above will be used to fit the Bogoliubov dispersion.
The article [50] also suggests that is possible to stabilize a vortex thanks to the combination of non-locality and nonlinear saturation.
3.7 Scattering on an optically induced defect: theory
In this section it will be discussed how an optical analogue of superfluidity can be observed in the present context of fluids of light in a propagating geometry.
One of the main feature of a superfluid behaviour is the suppression of the scattering in the flow around an obstacle.
In a normal fluid, the kinetic energy of the fluid is dissipated in the form of waves that are scattered from the obstacle. Instead in a superfluid exists a critical flow speed (as explained in 2.1) under which such waves are not allowed to be excited, and consequently the fluid passes the obstacle undisturbed. At the breakdown of superfluidity, i.e. at flows speeds close to the critical flow velocity, the turbulences manifests in the form of quantized vortex nucleation and this is commonly considered to be a hallmark signature of superfluidity [18].
The suppression of the friction felt by a physical defect has already been observed with exciton-polariton [11].
In this experiment an external potential is added to the NLSE using another laser beam with a radius way smaller than the other one.
i|∂A(¯r⊥, z)|
∂z = (− 1 2k0
∇2⊥+V0+g|A(¯r⊥, z)|2)A(¯r⊥, z) . (3.65)
The situation studied here is that of a fluid of light propagating at a finite speed and hitting a gaussian defect.
Using a Gaussian lasers profile such a defect can be described as a gaussian modula-tion of the linear dielectric constant:
δ(r⊥, z) =δmaxe−
r2
⊥
2σ2 , (3.66)
that is centered at r⊥ = 0 and with FWHMσ. This define the external potential V(r⊥, z) =−k0δ(r⊥,z)
2n20 .
The superfluidity condition is achieved when the dimensionless speed of light in the mediumv is smaller than the critical velocity, that correspond to the analogue speed of sound for a local nonlinearity while it is smaller in presence of non-locality. We recall the expression of the two speeds:
where the speed of the fluid v is considered as the relative angle φ outside of the medium, that is refracted inside of the medium following the Snell’s law.
Therefore the superfluidity condition for a local nonlinearity is:
φ <√ In such a case, since no states are any longer available for scattering at the frequency of the driving photon field, the photon scattering from the defect is inhibited and the photon fluid is able to flow without friction.
The idea for this experiment (and the simulations) comes from an article of I.Carusotto [5]. Here the simulations of the expected behaviour of the fluid in presence of a defect will be presented.
The figure 3.4 presents three different propagation regimes for the fluid at a fixed in-plane velocity v = 0.034, in presence of a constant gaussian defect while the incident background intensities is increased from left to right (the speed of sound increases).
Also here varying the incidence angle at a fixed light intensity the physics is the same, i.e it is possible to modify the speed of light in the transverse plane instead of the speed of sound.
Figure 3.4 a) shows the linear regime withvcs that occurs in the linear optics regime at very low incident intensities (or big angle): in this case, the linear interfer-ence of the incident and scattered light is responsible of the parabolic shape of the fringes.
As it will be shown later in a nonlinear medium similar fringes could be observed also because of self-phase modulation, but in this case they have a bigger visibility and are not parabolic for k⊥6= 0.
It is worth to notice, looking at the colour scale, that here the expected visibility of the scattering fringes is around 10 %.
Figure 3.4 b) shows a supersonic flow regime wherev > csand superflow is broken: a Mach–Cerenkov cone appears after the defect, with an aperture angle θ= arcsincvs. In the panel c) the fluid of light moves at a subsonic speed, and therefore it flows without friction, that means that there is no scattered light that interfere with the incident one, as is explained better in Fig. 3.5.
3.7 Scattering on an optically induced defect: theory 41
Figure 3.4. Long-distance asymptotic transverse profiles of the laser beam intensity hitting a cylindrical defect located at r⊥ = 0. Here the defect parameters are λσ
0 = 5 and
δmax
= −1.6×10−4. The flow velocity is the same v = 0.034 along the positive x-direction (the right-ward direction in the figure). The colour scale is normalized to the incident intensity. The Mach numbers for the three panels are: a) Linear regime v/cs=∞; b) Supersonic regimev/cs= 1.88; c) Superfluid regimev/cs= 0.86. Taken from [5].
Figure 3.5. Above are shown the transverse profiles of the laser beam intensity after a propagation distanceL/λ0= 4500 in the nonlinear medium. The parameters for a),b),c) are the same of figure 3.4. Below are shown the far field emission patterns for the same configurations as in the figure above in a logarithmic grey scale. The blue circle indicates the incident wavevectorkinc⊥ . Taken from [5].
From an optical point of view a suppressed scattering is associated to a suppressed radiation pressure acting on the defect [17].
Looking in the reciprocal space (Fig. 3.5, panel below) it is possible to observe all the range of k-vector acquired by the scattered light, i.e. a scattering ring around the range of k⊥inc proper of the beam.
The linear case (csv) is shown in panel (a). Here the scattering on the defect is responsible for a ring- shaped k-vector distribution, passing through the incident wave-vectork⊥inc (blue circle).
Increasing cs, i.e. going in the supersonic regime (panel (b) below), the ring is deformed close to the incident wave-vector kinc⊥ and a weaker copy of it appears sim-metrically to kinc⊥ . This is because in the supersonic regime the range of wave-vector that can be acquired in the scattering are limited.
In the superfluid regime (cs> v), shown in panel (c), the scattering is suppressed and therefore the ring disappears. Only a single peak at k⊥inc remains visible: the strong broadening of this peak is due to the expansion of the spot under the effect of the Kerr defocusing.